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The abstract homotopy theory is based on the observation that
analogues of much of the topological homotopy theory and simple
homotopy theory exist in many other categories (e.g. spaces over a
fixed base, groupoids, chain complexes, module categories).
Studying categorical versions of homotopy structure, such as
cylinders and path space constructions, enables not only a unified
development of many examples of known homotopy theories but also
reveals the inner working of the classical spatial theory. This
demonstrates the logical interdependence of properties (in
particular the existence of certain Kan fillers in associated
cubical sets) and results (Puppe sequences, Vogt's Iemma, Dold's
theorem on fibre homotopy equivalences, and homotopy coherence
theory).
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